New Game: Should I save the pawn?
This game is like a cousin of poisoned pawn. In it, you have a pawn that is under fire, and you must decide whether your best move is to defend the pawn or not (maybe another tactic is lurking in the position, or maybe defending the pawn ruins your position). The structure is like poisoned pawn, where you don't need to find the best move, just pick yes or no.
Best name I can come up with is Pawn Stocks, because you are choosing whether to invest in your pawn or "sell" it by letting it be captured.